Internal fe approximation of spaces of divergence-free functions in three-dimensional domains

SUMMARY The space of divergence-free vector functions with vanishing normal flux on the boundary is approximated by subspaces of finite elements having the same property. An easy way of generating basis functions in these subspaces is shown.

Nonstationary heat conduction in a stator

In this chapter we describe a method for computing the 3d nonstationary temperature field in the lamination pack of a stator of a synchronous or asynchronous (induction) motor with a centrifugal, meander or chamber ventilation (see [Křižek, Preiningerova]). The stator of a motor has quite a complicated geometrical form. Moreover, it consists of anisotropic materials which have very different heat conductivities, e.g., 332.8 [W/mK] for copper wires and 0.2 [W/mK] for their insulations. This causes big jumps in coefficients of the appropriate heat conduction equation, and is the main source of numerical difficulties in practical calculations.

The time-harmonic Maxwell equations

In this chapter we shall see that the solution of the time-harmonic Maxwell equations with real coefficients can be transformed to time independent partial differential equations with complex coefficients. Then we introduce a finite element approximation proposed in [Křižek, Neittaanmaki, 1989]. A similar technique is analyzed in [Křižek, Neittaanmaki, 1984b], [Monk, 1992a] (for fully time dependent problems see, e.g., [Monk 1992b,c]).

Steady-state radiation heat transfer problem

In Section 8.2, we shall see that the steady-state radiative heat transfer problem can be transformed to minimization of a smooth nonquadratic functional J over a convex and closed subset of a Banach space V. To this end we firstly shortly recall some basic definitions concerning differentiability of J, because these sometimes differ in the literature.

Magnetic potential of transformer window

We describe how to calculate the magnetic potential in the window of an ideal transformer. The knowledge of this potential is a starting point for the determination of some other quantities of practical importance (such as leakage field, overheating of windings, circular currents, additional losses, short circuit forces in windings, etc.).

Nonlinear anisotropic heat conduction in a transformer magnetic core

In this chapter we deal with a quasilinear elliptic problem whose classical formulation reads: Find \( u \in {C^1}\left( {\bar \Omega } \right) \) such that u|Ω ∈ C 2(Ω) and $$ - div\left( {A\left( { \cdot ,u} \right)grad\;u} \right) = f\quad in\;\Omega $$ (9.1) $$ u = \bar u\quad on\;{\Gamma _1} $$ (9.2) $$ \alpha u + {n^T}A\left( { \cdot ,u} \right)grad\;u = g\quad on\;{\Gamma _2} $$ (9.3) where Ω ∈ L, n = (n 1, ..., n d ) T is the outward unit normal to ∂Ω, d ∈ {1, 2, ...,}, Γ1 and Γ2 are relatively open sets in the boundary ∂Ω, \({\overline \Gamma _1} \cup {\overline \Gamma _2} = \partial \Omega ,\,{\Gamma _1} \cap {\Gamma _2} = \phi\), \( A = \left( {{a_{ij}}} \right)_{i,j = 1}^d \) is…

Mathematical modelling of physical phenomena

The word “physics” is derived from the Greek “fysis” which means nature. Physics investigates fundamental natural phenomena and thus physical knowledge has a very general character. This is also the reason why physics penetrates other areas including electrical engineering.

Postprocessing of a Finite Element Scheme with Linear Elements

In this contribution we first give a brief survey of postprocessing techniques for accelerating the convergence of finite element schemes for elliptic problems. We also generalize a local superconvergence technique recently analyzed by Křižek and Neittaanmaki ([20]) to a global technique. Finally, we show that it is possible to obtain O(h4) accuracy for the gradient in some cases when only linear elements are used. Numerical tests are presented.

On superconvergence techniques

A brief survey with a bibliography of superconvergence phenomena in finding a numerical solution of differential and integral equations is presented. A particular emphasis is laid on superconvergent schemes for elliptic problems in the plane employing the finite element method.

Superconvergence phenomenon in the finite element method arising from averaging gradients

We study a superconvergence phenomenon which can be obtained when solving a 2nd order elliptic problem by the usual linear elements. The averaged gradient is a piecewise linear continuous vector field, the value of which at any nodal point is an average of gradients of linear elements on triangles incident with this nodal point. The convergence rate of the averaged gradient to an exact gradient in theL 2-norm can locally be higher even by one than that of the original piecewise constant discrete gradient.

Methods for optimal shape design of electrical devices

Often the primary problem facing designers of structural systems is determining the shape of the structure. In spite of graphical work stations and modern software for analyzing the structure, finding the best geometry for the structure by “trial and error” is still a very tedious and timeconsuming task. The goal in optimal shape design (structural optimization, or redesign) is to computerize the design process and therefore shorten the time it takes to design new products or improve the existing design. Structural optimization is already used in many applications in industry. In general, however, structural optimization is just beginning to penetrate the industrial community. Integrating F…

On a global superconvergence of the gradient of linear triangular elements

Abstract We study a simple superconvergent scheme which recovers the gradient when solving a second-order elliptic problem in the plane by the usual linear elements. The recovered gradient globally approximates the true gradient even by one order of accuracy higher in the L 2 -norm than the piecewise constant gradient of the Ritz—Galerkin solution. A superconvergent approximation to the boundary flux is presented as well.

Stationary semiconductor equations

The behaviour of a semiconductor device is usually modelled by three coupled nonlinear partial differential equations of elliptic type. Such a system for the transport of mobile charge carriers was first introduced by Van Roosbroeck [Van Roosbroeck] in 1950. Nowadays there are many models which differ in their choice of unknowns, scales, various types of nonlinearities etc. (see, e.g., [Brezzi], [Groger], [Markowich], [Markowich, Ringhofer, Schmeiser], [Mock, 1972], [Polak, den Heijer, Schilders, Markowich], [Pospisek], [Pospisek, Segeth, Silhan], [Selberherr], [Sze], [Zlamal, 1986]).

Finite element analysis of varitional crimes for a quasilinear elliptic problem in 3D

We examine a finite element approximation of a quasilinear boundary value elliptic problem in a three-dimensional bounded convex domain with a smooth boundary. The domain is approximated by a polyhedron and a numerical integration is taken into account. We apply linear tetrahedral finite elements and prove the convergence of approximate solutions on polyhedral domains in the $W^1_2$ -norm to the true solution without any additional regularity assumptions.

Post‐processing of Gauss–Seidel iterations

Calculation of nonlinear stationary magnetic field

Currently, linear models of various physical fields can successfully be implemented numerically. Efficient numerical methods have been developed during last two or three decades and sufficiently capable computers are available. The situation is different with nonlinear models. There is no general numerical method for solving all nonlinear problems, and consequently every class of problems has to be investigated individually. The specific features of the class are taken into account in this process [Berger].

finite element methods

Two robot patch recovery methods with built-in field equations and boundary conditions superconvergence similarities in standard and mixed finite element methods on the FEM for the Navier-Stokes equations in the domains with corner singularities projections in finite element analysis and application element analysis method and superconvergence quadratic interpolation polynomials in vertices of strongly regular triangulations explicit error bounds for a nonconforming finite element method analysis of the average efficiency of an error estimator on the mesh for difference schemes of higher accuracy for the heat-conduction equation shape design sensitivity formulae approximated by means of a r…

Approximation of the Maxwell equations in anisotropic inhomogeneous media

Let Ω ∈ L be in ℝ 2. We consider the initial-boundary value problem $$ \begin{array}{l}rot\,E\left( {x,t} \right) + \mu \left( x \right)\frac{\partial }{{\partial t}}H\left( {x,t} \right) = J\left( {x,t} \right), \\\left( {x,t} \right) \in \Omega \, \times \,(0,T], \\curl\,H\left( {x,t} \right) - \varepsilon \left( {\frac{\partial }{{\partial t}}} \right)E\left( {x,t} \right) = k\left( {x,t} \right), \\n \wedge E\left( {x,t} \right) = 0, \\\left( {x,t} \right) \in \partial \Omega \, \times \,(0,T], \\\left( {E\left( {x,0} \right),H\left( {x,0} \right)} \right) = \left( {{E_0}\left( x \right),\,{H_0}\left( x \right)} \right), \\x \in \bar \Omega \\\end{array} $$ (13.1) .